Practical Guide to Understanding Implied Volatilities
A mathematical approach to determine and understand implied volatilities
- Implied volatility is a key component when determining the theoretical price of a European call option
- Unlike price, implied volatility cannot be instantaneously measured. It’s an average and needs to be estimated overtime
- The Black-Scholes-Merton (1973) option pricing formula is used to determine implied volatility (implied volatility is not an input parameter for the formula, but the result of an optimization procedure given that formula)
Let’s take a look at the famous option pricing formula offered by Black-Scholes-Merton (1973):
The different parameters of the formula have the following meaning:
Now let’s suppose we are given an option quote for a European call option C*. The implied volatility is simply the quantity that solves the aforementioned implicit equation .
The following equation represents the implied volatility given a market quote for an option. Our goal is to solve this equation in order to compute the implied volatility (unknown variable).
Mathematically speaking, there is no closed-form solution to this equation. We need to use numerical computing schemes such as Newton’s Method in order to estimate the correct solution. The Newton’s method iterates using the first derivative of the relevant function until a certain degree of precision is reached starting with an initial guess. To learn more about the Newton’s Method, feel free to check out this incredible tutorial.
It becomes apparent in the formula above (see denominator of equation 3–3) that we need to compute the partial derivative of the option pricing formula (equation 3–1) with respect to volatility. In options pricing theory, this metric is called Vega.
The derivation of Vega is as follows:
Now that we have the necessary numerical tools, we will use Python to demonstrate how implied volatilities are computed in the real world using Python in a future post.
Hilpisch, Y. (2015). Python for Finance: Analyze Big Financial Data